Resistivity logging systems and methods employing ratio signal set for inversion

ABSTRACT

Electromagnetic resistivity logging systems and methods yielding formation anisotropy and dip from a signal set that closely approximates the response of a idealized tool. One illustrative method embodiment derives from an azimuthally-sensitive tool&#39;s measurements a full set of orthogonal direct couplings (Vxx, Vyy, Vzz) and a cross-coupling sum (Vxz+Vzx) or (Vyz+Vzy). These values are converted into a signal set as a function of borehole position, the set including: a first signal representing a ratio between Vzz coupling components at different spacing distances, a second signal representing a ratio between Vxx and Vzz coupling components, a third signal representing a ratio between Vyy and Vzz coupling components, a fourth signal representing a ratio between Vxx and Vyy coupling components, and a fifth signal representing a ratio between a cross-coupling sum and a sum of the direct couplings. From this signal set, formation parameters can be accurately determined by inversion.

BACKGROUND

The basic principles and techniques for electromagnetic logging forearth formations are well known. For example, induction logging todetermine the resistivity (or its inverse, conductivity) of earthformations adjacent a borehole has long been a standard and importanttechnique in the search for and recovery of subterranean petroleumdeposits. In brief, a transmitter transmits an electromagnetic signalthat passes through formation materials around the borehole and inducesa signal in ore or more receivers. The amplitude and/or phase of thereceiver signals are influenced by the formation resistivity, enablingresistivity measurements to be made. The measured signal characteristicsand/or formation properties calculated therefrom are recorded as afunction of the tool's depth or position in the borehole, yielding aformation log that can be used by analysts.

Note, however, that the resistivity of a given formation may beisotropic (equal in all directions) or anisotropic (unequal in differentdirections). In electrically anisotropic formations, the anisotropy isgenerally attributable to fine layering during the sedimentary build-upof the formation. Hence, in a formation coordinate system oriented suchthat the x-y plane is parallel to the formation layers and the z axis isperpendicular to the formation layers, resistivities R_(X) and R_(Y) indirections x and y, respectively, tend to be the same, but resistivityR_(z) in the z direction is different. Thus, the resistivity in adirection parallel to the plane of the formation (i.e., the x-y plane)is often known as the horizontal resistivity, R_(H), and the resistivityin the direction perpendicular to the plane of the formation (i.e., thez direction) is often known as the vertical resistivity, R_(V). Theindex of anisotropy, is defined as η=[R_(V)/R_(H)]^(1/2).

As a further complication to measuring formation resistivity, boreholesare generally not perpendicular to formation beds. The angle between theaxis of the well bore and the orientation of the formation beds (asrepresented by a vector normal to the formation bed) has two components.These components are the dip angle and the strike angle. The dip angleis the angle between the borehole axis and the normal vector for theformation bed. The strike angle is the direction in which the boreholesaxis “leans away from” the normal vector. (These will be defined morerigorously in the detailed description.)

Electromagnetic resistivity logging measurements are a complex functionof formation resistivity, formation anisotropy, and the formation dipand strike angles, which may all be unknown. Moreover, engineers oftenrely on simplified models to interpret the measurements in a suitablyprompt manner. Logging tools that fail to account for each of theunknown parameters and differences between the model and the operationof the “real world” tool may provide measurement quality that is lessthan ideal. Conversely, tools that account for each of these factorswill provide improved resistivity measurements. Moreover, tools that areable to provide dip and strike measurements along with azimuthalorientation information, can be used for geosteering.

BRIEF DESCRIPTION OF THE DRAWINGS

Accordingly, there are disclosed herein resistivity logging systems andmethods employing a set of ratio-based signals as an inversion basis fordetermining formation parameters such as horizontal resistivity,anisotropy, dip, and strike angles. In the drawings:

FIG. 1 shows an illustrative logging while drilling environment.

FIG. 2 shows an illustrative wireline logging environment.

FIG. 3 shows a relationship between coordinate axes of a borehole and adipping formation bed.

FIG. 4 shows an orthogonal triad antenna arrangement for anelectromagnetic logging tool.

FIG. 5A shows angles for defining the orientation of a tilted antenna.

FIG. 5B shows azimuthal bins around a borehole circumference.

FIG. 6 is a block diagram of an illustrative electronics module for anelectromagnetic logging tool.

FIG. 7 shows an illustrative electromagnetic logging tool having tiltedtransmit and receive antennas.

FIGS. 8A-8D show alternative antenna configurations for anelectromagnetic logging tool.

FIGS. 9A-9D compare the phase and amplitude of uncalibrated andcalibrated signal coupling components.

FIGS. 10A-10B show the phase and amplitude of an illustrative robustsignal.

FIGS. 11A-11B show the phase and amplitude of illustrativeinstrumentation drift for the robust signal.

FIGS. 12A-12J show the illustrative phase and amplitude response of arobust signal set.

FIGS. 13A-13E show illustrative phase logs for the robust signal set.

FIG. 14 is a flowchart of an illustrative electromagnetic loggingmethod.

It should be understood, however, that the specific embodiments given inthe drawings and detailed description below do not limit the disclosure.On the contrary, they provide the foundation for one of ordinary skillto discern the alternative forms, equivalents, and other modificationsthat are encompassed in the scope of the appended claims.

DETAILED DESCRIPTION

The disclosed tool configurations and operations are best understood inthe context of the larger systems in which they operate. Accordingly, anillustrative logging while drilling (LWD) environment is shown inFIG. 1. A drilling platform 2 supports a derrick 4 having a travelingblock 6 for raising and lowering a drill string 8. A kelly 10 supportsthe drill string 8 as it is lowered through a rotary table 12. A drillbit 14 is driven by a downhole motor and/or rotation of the drill string8. As bit 14 rotates, it creates a borehole 16 that passes throughvarious formations 18. A pump 20 circulates drilling fluid through afeed pipe 22 to kelly 10, downhole through the interior of drill string8, through orifices in drill bit 14, back to the surface via the annulusaround drill string 8, and into a retention pit 24. The drilling fluidtransports cuttings from the borehole into the pit 24 and aids inmaintaining the borehole integrity.

An electromagnetic resistivity logging tool 26 is integrated into thebottom-hole assembly near the bit 14. As the bit extends the boreholethrough the formations, logging tool 26 collects measurements relatingto various formation properties as well as the tool orientation andposition and various other drilling conditions. The logging tool 26 maytake the form of a drill collar, i.e., a thick-walled tubular thatprovides weight and rigidity to aid the drilling process. A telemetrysub 28 may be included to transfer tool measurements to a surfacereceiver 30 and to receive commands from the surface receiver.

The tool orientation measurements may be performed using an azimuthalorientation indicator, which may include magnetometers, inclinometers,and/or accelerometers, though other sensor types such as gyroscopes canbe used. Most preferably, the orientation measurements are collectedusing both a 3-axis fluxgate magnetometer and a 3-axis accelerometer. Asis known in the art, the combination of those two sensor systems enablesthe measurement of the toolface, borehole inclination, and compassdirection of the borehole. The toolface and hole inclination angles arecalculated from the accelerometer sensor output. The magnetometer sensoroutputs are used to calculate the compass direction. With the toolface,the hole inclination, and the compass information, a tool in accordancewith the present disclosure can be used to steer the bit to thedesirable bed.

At various times during the drilling process, the drill string 8 may beremoved from the borehole as shown in FIG. 2. Once the drill string hasbeen removed, logging operations can be conducted using a wirelinelogging tool 34, i.e., a sensing instrument sonde suspended by a cable42 having conductors for transporting power to the tool and transportingtelemetry from the tool to the surface. The illustrated sonde includes aresistivity logging tool 34 having centralizing arms 36 that center thetool within the borehole as the tool is pulled uphole. A loggingfacility 44 collects measurements from the logging tool 34, and includescomputing facilities for processing and storing the measurementsgathered by the logging tool.

FIG. 1 shows that the formations 18 are not perpendicular to theborehole, which may occur naturally or may be due to directionaldrilling operations. The borehole has a coordinate system 50 defined inaccordance with the borehole's long axis (the z axis) and the north side(or alternatively, the high side) of the hole (the x-axis). Theformations 18, when characterized as a plane, have a coordinate system51 defined in accordance with the normal to the plane (the z″ axis) andthe direction of steepest descent (the x″-axis). As shown in FIG. 3, thetwo coordinate systems are related by two rotations. Beginning with theborehole's coordinate system (x,y,z), a first rotation of angle γ ismade about the z axis. The resulting coordinate system is denoted(x′,y′,z′). Angle γ is the relative strike angle, which indicates thedirection of the formation dip relative to the borehole's coordinatesystem. A second rotation of angle α is then made about the y′ axis.This aligns the borehole coordinate system with the formation coordinatesystem. Angle α is the relative dip angle, which is the slope angle ofthe beds relative to the long axis of the borehole.

The vertical resistivity is generally defined to be the resistivity asmeasured perpendicular to the plane of the formation, and the horizontalresistivity is the resistivity as measured within the plane of theformation. Determination of each of these parameters (dip angle, strikeangle, vertical resistivity, and horizontal resistivity) is desirable.

FIG. 4 shows a hypothetical antenna configuration for a multi-componentelectromagnetic resistivity logging tool. (The electromagneticresistivity logging tool may be embodied as a wireline tool and as alogging while drilling tool.) A triad of transmitter coils T_(X), T_(Y)and T_(Z), each oriented along a respective axis, is provided. At leastone triad of similarly oriented receiver coils R_(X), R_(Y), and R_(Z)is also provided at some distance from the transmitter triad. Moran andGianzero, in “Effects of Formation Anisotropy on Resistivity LoggingMeasurements” Geophysics, Vol. 44, No. 7, p. 1266 (1979), noted that themagnetic field h in the receiver coils can be represented in terms ofthe magnetic moments m at the transmitters and a coupling matrix C:h=mC  (1)In express form, equation (1) is:

$\begin{matrix}{{\begin{bmatrix}H_{x} & H_{y} & H_{z}\end{bmatrix} = {\begin{bmatrix}M_{x} & M_{y} & M_{z}\end{bmatrix}\begin{bmatrix}C_{xx} & C_{xy} & C_{xz} \\C_{yx} & C_{yy} & C_{zz} \\C_{zx} & C_{zy} & C_{zz}\end{bmatrix}}},} & (2)\end{matrix}$where M_(X), M_(Y), and M_(Z) are the magnetic moments (proportional totransmit signal strength) created by transmitters T_(X), T_(Y), andT_(Z), respectively. H_(X), H_(Y), H_(Z) are the magnetic fields(proportional to receive signal strength) at the receiver antennasR_(X), R_(Y), and R_(Z), respectively.

In the antenna configuration of FIG. 4, if each transmitter is fired inturn, and signal measurements are made at each receiver in response toeach firing, nine signal measurements are obtained. These ninemeasurements enable the determination of a complete coupling matrix C.(C_(IJ)=a_(IJ)V_(IJ), where I is the index for transmitter T_(X), T_(Y),or T_(Z), J is the index for receiver R_(X), R_(Y), or R_(Z), a_(IJ) isa constant determined by the tool design, and V_(IJ) is a complex valuerepresenting the signal amplitude and phase shift measured by receiver Jin response to the firing of transmitter I.) Knowledge of the completecoupling matrix enables the determination of dip angle, strike angle,vertical resistivity, and horizontal resistivity. A number of techniquesmay be used to determine these parameters. For example, dip and strikeangle may be determined from coupling matrix values as explained by LiGao and Stanley Gianzero, U.S. Pat. No. 6,727,706 “Virtual Steering ofInduction Tool for Determination of Formation Dip Angle”. Given theseangles, vertical and horizontal resistivity can be determined inaccordance with equations provided by Michael Bittar, U.S. Pat. No.7,019,528 “Electromagnetic Wave Resistivity Tool Having a Tilted Antennafor Geosteering within a Desired Payzone”. Alternatively, a simultaneoussolution for these parameters may be found as described in the Bittarreference.

FIG. 5A shows two angles that may be used to specify the orientation ofa tilted coil antenna. The tilted coil antenna may be considered asresiding in a plane having a normal vector. Tilt angle θ is the anglebetween the longitudinal axis of the tool and the normal vector. Azimuthangle β is the angle between the projection of the normal vector in theX-Y plane and the tool scribe line. Alternatively, in the downholecontext, azimuthal angle β may represent the angle between projection ofthe normal vector in the X-Y plane and the x-axis of the boreholecoordinate system. FIG. 5B shows a division of the boreholecircumference into n bins, each bin corresponding to a range ofazimuthal angle values. A representative (e.g., average) azimuthal angleis associated with each bin. Tilted antenna measurements may beassociated with the bin containing the azimuthal angle for that antenna,the angle (and corresponding bin) changing as the tool rotates.

It is noted that three transmitter antenna orientations and threereceiver antenna orientations are employed in the antenna configurationof FIG. 4. It has been discovered that when tool rotation is exploited,it is possible to determine the full coupling matrix with only onetransmitter and two receiver antenna orientations (or equivalently, onereceiver and two transmitter antenna orientations). Of course, moretransmitter and/or receiver antennas can be employed and may be helpfulfor producing more robust measurements as described below.

Before considering various tools having specific antenna configurations,the electronics common to each tool are described. FIG. 6 shows afunctional block diagram of the electronics for a resistivity tool. Theelectronics include a control module 602 that is coupled to an analogswitch 604. Analog switch 604 is configured to drive any one of thetransmitter coils T₁, T₂, T₃, T₄ with an alternating current (AC) signalfrom a signal source 606. In at least some embodiments, the signalsource provides radio frequency signals. The control module 602preferably selects a transmitter coil, pauses long enough for transientsto die out, then signals data storage/transmit module 610 to accept anamplitude and phase sample of the signals measured by each of thereceiver coils. The control module 602 preferably repeats this processsequentially for each of the transmitters. The amplitude and phase shiftvalues are provided by amplitude and phase shift detector 608 which iscoupled to each of the receiver coils R₁-R₄ for this purpose.

Control module 602 may process the amplitude and phase shiftmeasurements to obtain compensated measurements and/or measurementaverages. In addition to being stored in memory downhole, the raw,compensated, or averaged measurements may be transmitted to the surfacefor processing to determine coupling matrix elements, dip and strikeangles, vertical and horizontal resistivity, and other information suchas (i) distance to nearest bed boundary, (ii) direction of nearest bedboundary, and (iii) resistivity of any nearby adjacent beds.Alternatively, all or some of this processing can be performed downholeand the results may be communicated to the surface. The datastorage/transmitter module 610 may be coupled to telemetry unit 28(FIG. 1) to transmit signal measurements or processing results to thesurface. Telemetry unit 28 can use any of several known techniques fortransmitting information to the surface, including but not limited to(1) mud pressure pulse; (2) hard-wire connection; (3) acoustic wave; and(4) electromagnetic waves.

FIG. 7 shows an electromagnetic resistivity logging tool 702 having onlytwo receiver antenna orientations. The tool 702 is provided with one ormore regions 706 of reduced diameter. A wire coil 704 is placed in theregion 706 and in some embodiments is spaced away from the surface ofsubassembly 702 by a constant distance. To mechanically support andprotect the coil 704, a non-conductive filler material (not shown) suchas epoxy, rubber, or ceramic may be used in the reduced diameter regions706. Coil 704 is a transmitter coil, and coils 710 and 712 are receivingcoils, though these roles can be reversed in view of the principle ofreciprocity. In operation, transmitter coil 704 transmits aninterrogating electromagnetic signal which propagates through theborehole and surrounding formation. Receiver coils 710, 712 detect theinterrogating electromagnetic signal and provide a measure of theelectromagnetic signal's amplitude attenuation and phase shift. Fordifferential measurements additional receiver coils parallel to coils710, 712 may be provided at an axially-spaced distance (see, e.g., FIG.8). From the absolute or differential amplitude attenuation and phaseshift measurements, the coupling matrix components can be determined andused as the basis for determining formation parameters and as the basisfor geosteering.

In some embodiments, the transmitter coil 704 is spaced approximately 30inches from the receiver coils 710, 712. The additional receiver coilscould be positioned approximately 8 inches further from the transmittercoil. The transmitter and receiver coils may comprise as little as oneloop of wire, although more loops may provide additional signal power.The distance between the coils and the tool surface is preferably in therange from 1/16 inch to ¾ inch, but may be larger. Transmitter coil 704and receiver coil 712 may each have a tilt angle of about 45° andaligned with the same azimuth angle, while receiver coil 710 may have atilt angle of about 45° and an azimuth 180° apart from receiver coil 712(or equivalently, a tilt angle of minus 45° at the same azimuth angle asreceiver coil 712).

The signal measured by a tilted receiver in response to the firing of atilted transmitter can be expressed in terms of the signals V_(IJ) thatwould be measured by the tool of FIG. 4. When both transmitter andreceiver coils are oriented at the same azimuth angle β, the tiltedreceiver signal Y_(R) is

$\begin{matrix}{{V_{R}(\beta)} = {{\begin{bmatrix}{\sin\;\theta_{T}\cos\;\beta} \\{\sin\;\theta_{T}\sin\;\beta} \\{\cos\;\theta_{T}}\end{bmatrix}^{T}\begin{bmatrix}V_{xx} & V_{yx} & V_{zx} \\V_{xy} & V_{yy} & V_{yz} \\V_{xz} & V_{yz} & V_{zz}\end{bmatrix}}\begin{bmatrix}{\sin\;\theta_{R}\cos\;\beta} \\{\sin\;\theta_{R}\sin\;\beta} \\{\cos\;\theta_{R}}\end{bmatrix}}} & (3)\end{matrix}$where θ_(T) is the tilt angle of the transmitter and θ_(R) is the tiltangle of the receiver. In written-out form, the received signal is:

$\begin{matrix}{\mspace{79mu}{\begin{matrix}{{V_{R}(\beta)} = {\left\lfloor {{\left( {\frac{C_{xx}}{2} - \frac{C_{yy}}{2}} \right)\cos\; 2\beta} + {\left( \frac{C_{yx} + C_{xy}}{2} \right)\sin\; 2\beta}} \right\rfloor +}} \\{\left\lbrack {{\left( {C_{zx} + C_{xz}} \right)\cos\;\beta} + {\left( {C_{zy} + C_{yz}} \right)\sin\;\beta}} \right\rbrack +} \\{\left( {C_{zz} + \frac{C_{xx}}{2} + \frac{C_{yy}}{2}} \right)} \\{= {{V_{double}(\beta)} + {V_{single}(\beta)} + V_{const}}}\end{matrix}\mspace{20mu}{{meaning}\mspace{14mu}{that}}}} & (4) \\{\mspace{79mu}\left\{ {\begin{matrix}{{V_{double}(\beta)} = {{\left( {\frac{C_{xx}}{2} - \frac{C_{yy}}{2}} \right)\cos\; 2\beta} + {\left( \frac{C_{yx} + C_{xy}}{2} \right)\sin\; 2\beta}}} \\{{V_{single}(\beta)} = {{\left( {C_{zx} + C_{xz}} \right)\cos\;\beta} + {\left( {C_{zy} + C_{yz}} \right)\sin\;\beta}}} \\{V_{const} = {C_{zz} + \frac{C_{xx}}{2} + \frac{C_{yy}}{2}}}\end{matrix}\mspace{20mu}{where}} \right.} & (5) \\\left\{ \begin{matrix}{{C_{xx} = {V_{xx}\sin\;\theta_{t}\sin\;\theta_{r}}};} & {{C_{yx} = {V_{yx}\sin\;\theta_{t}\sin\;\theta_{r}}};} & {{C_{zx} = {V_{zx}\cos\;\theta_{t}\sin\;\theta_{r}}};} \\{{C_{xy} = {V_{xy}\sin\;\theta_{t}\sin\;\theta_{r}}};} & {{C_{yy} = {V_{yy}\sin\;\theta_{t}\sin\;\theta_{r}}};} & {{C_{zy} = {V_{zy}\cos\;\theta_{t}\sin\;\theta_{r}}};} \\{{C_{xz} = {V_{xz}\sin\;\theta_{t}\cos\;\theta_{r}}};} & {{C_{yz} = {V_{yz}\sin\;\theta_{t}\cos\;\theta_{r}}};} & {{C_{zz} = {V_{zz}\cos\;\theta_{t}\cos\;\theta_{r}}};}\end{matrix} \right. & (6)\end{matrix}$

Sinusoidal curve fitting may be applied to the received signal toextract the (summed) coefficients in equation (5). The measurements of asecond tilted receiver's response to the tilted transmitter provides anadditional set of measurements that enables the individual C_(IJ) (orequivalently, the V_(IJ)) values to be obtained. (Note that in mostcases V_(xy) may be assumed equal to V_(yx), but the same is not truefor the other cross components.) As an example, take θ₁=θ_(r2)=θ_(c) andθ_(r1)=−θ_(r2), with the receivers R1 and R2 collocated at a distance d₁from the transmitter The zz coupling component with can be written as

$\begin{matrix}{{V_{zz}\left( d_{1} \right)} = \frac{V_{r\; 1\_\;{const}} + V_{r\; 2\_\;{const}}}{2\cos^{2}\theta_{c}}} & (7)\end{matrix}$where V_(r1) _(_) _(const) is the constant complex voltage V_(const)from equation (4) associated with receiver R1, and V_(r2) _(_) _(const)is the corresponding value for receiver R2. Along similar lines, the xxand yy components can be written

$\begin{matrix}{{V_{xx}\left( d_{1} \right)} = \frac{\left( {V_{r\; 1\_\;{const}} - V_{r\; 2\_\;{const}}} \right) + \left( {V_{r\; 1\_\;{double}\;{\_\cos}} - V_{r\; 2\_\;{double}\;{\_\cos}}} \right)}{2\sin^{2}\theta_{c}}} & \left( {7b} \right) \\{{V_{yy}\left( d_{1} \right)} = \frac{\left( {V_{r\; 1\_\;{const}} - V_{r\; 2\_\;{const}}} \right) + \left( {V_{r\; 1\_\;{double}\;{\_\cos}} - V_{r\; 2\_\;{double}\;{\_\cos}}} \right)}{2\sin^{2}\theta_{c}}} & \left( {7c} \right)\end{matrix}$The cross components can be written:

$\begin{matrix}{{V_{xy}\left( d_{1} \right)} = {{V_{yx}\left( d_{1} \right)} = \frac{V_{r\; 1\_\;{double}\;{\_\sin}} - V_{r\; 2\_\;{double}\;{\_\sin}}}{2\sin^{2}\theta_{c}}}} & \left( {8a} \right) \\{{V_{yz}\left( d_{1} \right)} = \frac{V_{r\; 1\_\;{single}\;{\_\sin}} + V_{r\; 2\_\;{single}\;{\_\sin}}}{2\cos\;\theta_{c}\sin\;\theta_{c}}} & \left( {8b} \right) \\{{V_{zy}\left( d_{1} \right)} = \frac{V_{r\; 1\_\;{single}\;{\_\sin}} - V_{r\; 2\_\;{single}\;{\_\sin}}}{2\cos\;\theta_{c}\sin\;\theta_{c}}} & \left( {8c} \right) \\{{V_{xz}\left( d_{1} \right)} = \frac{V_{r\; 1\_\;{single}\;{\_\cos}} - V_{r\; 2\_\;{single}\;{\_\cos}}}{2\cos\;\theta_{c}\sin\;\theta_{c}}} & \left( {8d} \right) \\{{V_{zx}\left( d_{1} \right)} = \frac{V_{r\; 1\_\;{single}\;{\_\cos}} - V_{r\; 2\_\;{single}\;{\_\cos}}}{2\cos\;\theta_{c}\sin\;\theta_{c}}} & \left( {8e} \right)\end{matrix}$

Other techniques for deriving the coupling components from the receivedsignal measurements are known and may be used. See, e.g., WO 2008/076130“Antenna coupling component measurement tool having a rotating antennaconfiguration” and WO 2011/129828 “Processing and Geosteering with aRotating Tool”.

To provide more robust measurements, additional transmitters and/orreceivers may be included on the tool as indicated in FIGS. 8A-8D. FIG.8A shows a tool having a first set of oppositely tilted receiverantennas (Rup1, Rup2, with respective skew angles −θc and +θc) at adistance d1 from a tilted transmitter antenna (Tup1 with skew angle +θc)and a second set of oppositely tilted receiver antennas (Rdn1, Rdn2 at−θc and +θc) at a distance d2 from tilted transmitter antenna Tup1. Theadditional set of receiver antennas enables cancelation of the mandreleffect as explained further below. The illustrated tool further includesa second tilted transmitter antenna (Tdn1 at +θc) positioned at distanced1 from the second set of receiver antennas and distance d2 from thefirst set of receiver antennas. The additional transmitter antennaenables compensation of temperature effects in the receiver electronicsas explained further below.

FIG. 8B shows an alternative antenna configuration in which theadditional transmitter antenna is skewed in an opposite direction fromthe first transmitter antenna. FIG. 8C shows an antenna configurationwith two sets of oppositely-tilted transmitter antennas (±θc) and asingle tilted receiver antenna at distance d1 and a single receiverantenna at distance d2, the two receiver antennas being parallel (+θc).FIG. 8D is similar, but has the two receiver antennas skewed in oppositedirections. Yet another antenna configuration would include two sets ofoppositely-tilted transmitter antennas together with two sets ofoppositely-tilted receiver antennas. It is further noted that thereceiver antennas are shown as being positioned between the transmitterantennas, but this is not a requirement, as some tool embodiments mayhave the transmitter antennas positioned between the receiver antennas.

Given the illustrative antenna configurations, the tool measurements maybe combined as outlined below to provide more robust values, i.e.,measurements that are insensitive to environmental effects (e.g.,temperature, pressure, and eccentricity) and that compensate for toolnon-idealities such as the presence of a conductive tool mandrel whenthe models assume point dipoles. As one step in this direction, the toolmay acquire measurements with a second set of receivers at a distance d2from the transmitter (see, e.g., FIG. 8A). The ratio (hereafter termed“Signal 1” or S1):S1=V _(zz)(d ₁)/V _(zz)(d ₂)  (9)has been found to significantly reduce sensitivity to the mandreleffect, and it serves as a good indication of formation resistivity.Signal 1 can be calibrated by means of an air-hang measurement in whichthe tool is suspended sufficiently far from any conductive or partiallyconductive materials (e.g., 20 feet in the air) and the received signalresponses noted. Representing the air-hang measurements with an “air”superscript, the calibrated signal is:

$\begin{matrix}{S_{1}^{cal} = {{\frac{V_{zz}\left( d_{1} \right)}{V_{zz}\left( d_{2} \right)}/\frac{V_{zz}^{air}\left( d_{1} \right)}{V_{zz}^{air}\left( d_{2} \right)}} = {\frac{V_{zz}\left( d_{1} \right)}{V_{zz}^{air}\left( d_{1} \right)}/\frac{V_{zz}\left( d_{2} \right)}{V_{zz}^{air}\left( d_{2} \right)}}}} & (10)\end{matrix}$The second expression above simply indicates that the calibration can beequivalently performed on a component by component basis.

Taking as an example the antenna configuration of FIG. 8A with a mandreldiameter of 4.0″, an antenna coil diameter of 4.5″ (as measured byprojecting the coil onto the x-y plane), a first transmit-receiveantenna spacing of d1=28″ and a second transmit-receive antenna spacingof d2=36″, and an operating signal frequency of 500 kHz, a simulationwas performed to illustrate the effects of the mandrel on the measuredsignals given by equations (9) and (10). As the signals arecomplex-valued, they are graphed in FIG. 9 in terms of their amplitudeand phase angle.

FIGS. 9A and 9B show the phase angle and amplitude of Signal 1 (equation9) as a function of formation resistivity and skew angle. Forcomparison, the expected signal derived from a point-dipole model isalso shown. FIGS. 9C and 9D show the phase angle and amplitude of thecalibrated Signal 1 (equation 10) as a function of formation resistivityand skew angle. The phase angle curves overlap so well as to beindistinguishable. FIG. 9B, however, indicates a significant mandreleffect when tools having low antenna skew angles measure formations withhigher resistivities. The calibrated signals, however, adequatelycorrect for this effect and bring the curves largely into alignment withthe point dipole model. As the slope of the signal curve flattens outfor high resistivities, the formation resistivity calculation may besensitive to small errors in this region.

To improve resistance to temperature effects, compensated measurementscan be used. Such compensation techniques are known (at least withrespect to tools using coaxial antennas), and they combine themeasurements extracted from the receivers' responses to the firstantenna with the receivers' responses to the second antenna. Forexample, denoting the calibrated signal measurement derived fromtransmitter Tup1 (FIG. 8B) as S₁ ^(cal) (T_(up1)) and the calibratedsignal measurement derived from Tdn1 as S₁ ^(cal) (T_(dn1)), thecompensated signal measurement can be expressed asS ₁ ^(comp)=√{square root over (S ₁ ^(cal)(T _(up1))·S ₁ ^(cal)(T_(dn1)))}  (11)An alternative compensation approach is to simply average the twocalibrated measurements. Depending on the antenna configuration it maybe desirable to precede this compensation calculation with depthshifting and/or azimuth reversal of the calibrated signal measurementsto ensure that the measurements collected using the different antennasare all associated with the same region of the formation. The additionalmeasurements can also improve signal to noise ratio.

Returning to a single-transmitter analysis, we now consider additionaltool signals. As with Signal 1, it is helpful to normalize the othercoupling components. Unfortunately, the distance effect on the xxcoupling component measurements is different than the effect on the zzcoupling component measurements, making a different approach desirable.The ratio (hereafter termed “Signal 2” or “S2”):S2(d)=V _(xx)(d)/V _(zz)(d)  (12)has been found to compensate for the mandrel effect and produce a betterapproximation of a point-dipole tool response. It can be calibrated andcompensated in a similar fashion as Signal 1. The responses of thesecond set of receivers can also be taken into account with ageometrical average, yielding a combined S2:S ₂ ^(combined)=√{square root over (S ₂(d ₁)·S ₂(d ₂))}  (13)which can also be calibrated and compensated as described previously.FIGS. 10A and 10B show the phase and amplitude of the combined,uncalibrated, uncompensated S2 as a function of resistivity and skewangle. (Because of the denominator in equation 7b, a zero skew angle isnot included.) A point-dipole tool model response is also shown. Forresistivities above about 0.2 Ωm, the match to the point-dipole model isquite good. A similar response is expected for Signal 3, which isdefined:S3(d)=V _(yy)(d)/V _(zz)(d)  (14)withS ₃ ^(combined)=√{square root over (S ₃(d ₁)·S ₃(d ₂))}  (15)

In a similar vein, a fourth signal may be defined:S4(d)=V _(xx)(d)/V _(yy)(d)  (16)withS ₄ ^(combined)=√{square root over (S ₄(d ₁)·S ₄(d ₂))}  (17)It is expected that an approximate skew angle of 45° would offer thebest noise immunity as it provides antenna responses with roughly equalsignal responses from the xx and zz components. We note that signal S4can be defined using the inverse ratio with equally effective results.

FIG. 11 illustrates the results of a cool-down test, in which a loggingtool is heated to 300° C. before being suspended in the air and allowedto cool to ambient temperature. FIGS. 11A and 11B show the phase andamplitude of the calibrated (but not combined or compensated) S2 signalas a function of temperature. The signal variation is approximately±0.1° of phase and ±2% of amplitude. With compensated measurements, thisvariation is essentially eliminated.

Though the foregoing four signals have each been defined in terms of aratio between two components, the definition of robust signals need notbe so limited. A fifth contemplated signal is defined as:

$\begin{matrix}{{S_{5}(d)} = \frac{{V_{xz}(d)} + {V_{zx}(d)} + {V_{const}(d)}}{V_{const}(d)}} & (18)\end{matrix}$The fifth signal is expressible as

$S_{5} = {1 + {\frac{V_{x\; z} + V_{z\; x}}{{\frac{1}{2}V_{x\; x}} + {\frac{1}{2}V_{y\; y}} + V_{z\; z}}.}}$The fifth signal offers an enhanced sensitivity to relative dip angle.As with the other signals, combined, calibrated, and compensatedversions of the fifth signal can be determined. With this set of robustmeasurement signals, one can expect to compute very reliable inversionresults.

The ratio, calibration, and compensation techniques that have beenapplied to the extracted coupling components can also be applied to thereceiver signals. Thus, for example, an calibrated azimuthal resistivitysignal can be expressed

$\begin{matrix}{{V_{Res}^{Calibrated}(\beta)} = \frac{\frac{{V_{{Rdn}\; 1}(\beta)} - {V_{{Rdn}\; 2}(\beta)}}{{V_{{Rup}\; 1}(\beta)} - {V_{{Rup}\; 2}(\beta)}}}{\frac{V_{{Rdn}\; 1\_\;{const}}^{air} - V_{{Rdn}\; 2\_\;{const}}^{air}}{V_{{Rup}\; 1\_\;{const}}^{air} - V_{{Rup}\; 2\_\;{const}}^{air}}}} & (19)\end{matrix}$or a geosteering signal based on operation of a first transmitter Tup1can be expressed

$\begin{matrix}{{V_{Geo}^{T_{{up}\; 1}}\left( {\beta,d_{1}} \right)} = \frac{{V_{R_{{up}\; 2}}\left( {\beta,d_{1}} \right)} + {V_{R_{{up}\; 1}}\left( {\beta,d_{1}} \right)}}{{V_{R_{{up}\; 2}}\left( {\beta,d_{1}} \right)} - {V_{R_{{up}\; 1}}\left( {\beta,d_{1}} \right)}}} & (20)\end{matrix}$and a combined geosteering signal can be expressed:V _(Geo) ^(T) ^(up1-) ^(comb)(β)=√{square root over (V _(Geo) ^(T)^(up1) (β,d ₁)·V _(Geo) ^(T) ^(up1) (β,d ₂))}  (21)

Eq. (17) to Eq. (19) illustrate general compensation methods that applyto azimuthal measurements of MWD/LWD tools with tilted antenna systems.These signals can be used to determine formation parameters, such asformation resistivity, formation anisotropy, formation relative dipangle, etc. In addition, due to the cancellation of mandrel andtemperature effects, these signals can be also used for look-aheadtools.

It is noted that signals S1-S5 are calculated from the couplingcomponents, and can be determined from these components regardless ofhow the coupling components were derived from the tool measurements.Among other things, this observation indicates that the principlesdisclosed herein can be applied to the measurements of any antennaconfiguration sufficient to determine the coupling components (includingthat of FIG. 4) and regardless of whether the tool is embodied inwireline or LWD form. It is further noted that while each of thecombination, calibration, and compensation operations described abovecan contribute to improving measurement accuracy, each of theseoperations is optional. The order in which the chosen operations areapplied is largely a matter of convenience and can be varied withoutsignificantly impacting their potential benefits. Some or all of thecombination, calibration, and/or compensation operations can beperformed on the extracted coupling components before or after the ratiocalculations that yield signals S1-S5, and in at least some cases, theycould be performed on the (azimuthally dependent) received signalmeasurements before the extraction of the coupling components.

We have found that signal set S1-S4 serves as an excellent set of inputsfrom which formation parameters such as horizontal resistivity,anisotropy, dip angle, and strike angle can be derived. Accuracy isimproved with the use of additional signals such as signal S5. FIGS.12A-12E illustrate the phases of signals S1-S5 and FIGS. 12F-12Hillustrate the amplitudes of signals S1-S5 for different resistivities,anisotropies, and dip angles in a homogeneous formation. The signals'sensitivities to each of these parameters is evident from an inspectionof these figures, and the calibration, combination, and compensationtechniques outlined previously do not inhibit this sensitivity, as hasbeen found to be the case for other noise reduction techniques.

Without limiting the manner in which the signal set is employed toderive the formation parameters, we note that the S1 signal closelyrelates to the operation of a conventional logging tool and indeed, canbe converted to a conventional resistivity signal. Signal S2 and S3 canbe used to determine resistive anisotropy of the formation. Signal S4captures the divergence of the xx and yy coupling components andprovides a useful sensitivity to dip angle. Signal S5 relates thecross-coupling components to the direct coupling components and servesto speed the inversion with its unique sensitivity to the formationparameters. Performing inversion on the set of signals S1-S4 or S1-S5yields a robust estimate of formation parameters.

FIGS. 13A-13E illustrate the set of signals S1-S5 derived from a set ofreal-world measurements (as provided in FIG. 14) by a tool having theantenna configuration of FIG. 8B. These signal logs may be printed,displayed on a computer screen, or otherwise made tangible for a user tostudy and analyze. The signal logs show signal phase as a function ofmeasured depth (i.e., position along the borehole). Inversion wasperformed on signals S1-S5 using a Levenberg-Marquardt technique with aOD inversion code for a point-dipole model yielded the predicted signalsset indicated by the broken lines in FIG. 13. A excellent match betweenthe derived (“raw”) signals and the predicted (“sim”) signals can beobserved. The obtained parameters also match what the petrophysicistsknow about this well from other sources.

FIG. 14 is a flowchart of an illustrative tilted antenna logging methodwhich may be performed by a downhole controller, by a surface computingfacility that receives measurements from the tool, or performedcooperatively by both. In block 802 an initial transmitter is selected.In block 804, the selected transmitter is fired, and the amplitude andphase of each receiver's response is measured. The tool's position andorientation are also captured and used to associate the receiverresponse measurements with an measurement bin. (Because the boreholewall is conceptually divided into a grid, each bin has both an angularextent and an axial extent.) In block 806, the latest measurements areused to update the average response for each receiver for the given bin.

In block 808, a test is made to determine whether additionalmeasurements are needed or will be forthcoming at the current boreholeposition. For example, in tools having multiple transmitters, it isdesired to have measurements from each transmitter. Other reasons forneeding additional measurements include having a desired number ofmeasurements within each measurement bin before additional processing isperformed, or having at least a given number of azimuthally differentmeasurements before additional processing is performed. If additionalmeasurements at the current position are expected, the additionalprocessing may be postponed until all the relevant measurements havebeen collected. The logging process then proceeds with the selection ofthe next transmitter in block 809 and blocks 804-809 are repeated untilsufficient measurements have been achieved for the current boreholeposition.

Once a sufficient number of measurements have been obtained at a givenposition in the borehole, the method continues with block 810, where theorthogonal antenna couplings are extracted from theazimuthally-dependent measurements collected at the current boreholeposition. This may be done in accordance with the equations (3)-(8)given above, or by any suitable method including a least squaressolution to a linear system of equations such as that disclosed in WO2008/076130 “Antenna coupling component measurement tool having arotating antenna configuration”. Certain antenna configurations (e.g.,those using orthogonal triads) may yield such measurements directly.

In block 812, signals S1-S5 are derived from the orthogonal componentsas described above. There may be a set of such signals for each ofmultiple transmit-receive antenna pairings, which can be subjected to acombination operation (to combine measurements by receivers at differentdistances) and/or a compensation operation (to combine measurementsobtained in response to different transmitters) to yield more accuratesignals S1-S5 in optional block 814. An optional calibration operationmay also be applied in block 814.

In block 816, an initial estimate of the formation parameters is made.This estimate can be based on default values, previous results, orrandomly generated. The contemplated formation parameters includehorizontal resistivity, anisotropy, dip angle, and strike, but otherparameters can be employed. In block 818, a predicted set of signalsS1-S5 is generated from a model based on the estimated formationparameter values. In block 820, the predicted signal set is compared tothe set of signals derived in blocks 812-814. If there is not anadequate match, the estimated values are updated in block 821 inaccordance with a Levenberg-Marquardt technique, a Gauss-Newtontechnique, or other numerical solution technique. Blocks 818-821 arerepeated until the predicted set of signals converges to the derivedset. Then, in optional block 822, a real-time log displaying one or moreof the formation parameters as a function of position is updated withthe newly determined parameter values. The log associates the calculatedvalues with a depth or axial position within the borehole.

In block 824 a check is made to determine if logging information isavailable (or will become available) for additional positions within theborehole. If so, the process begins again with block 802. Otherwise, theprocess terminates.

Numerous variations and modifications will become apparent to thoseskilled in the art once the above disclosure is fully appreciated. Forexample, the foregoing disclosure describes numerous antennaconfigurations in the context of a logging while drilling tool, suchantenna configurations can also be readily applied to wireline loggingtools. Furthermore, the principle of reciprocity can be applied toobtain equivalent measurements while exchanging each antenna's role as atransmitter or receiver. It is intended that the following claims beinterpreted to embrace all such variations and modifications.

What is claimed is:
 1. A resistivity logging method that comprises:obtaining signal measurements collected by an azimuthally sensitiveelectromagnetic logging tool as a function of position in a borehole,the tool having at least two spacing distances (d1, d2) between transmitand receive antennas; deriving from the signal measurements: orthogonaldirect couplings (Vxx, Vyy, Vzz) and a cross-coupling sum (Vxz+Vzx) or(Vyz+Vzy); generating a set of signals as a function of position in theborehole, the set including: a first signal representing a ratio betweenVzz coupling components at different spacing distances, a second signalrepresenting a ratio between Vxx and Vzz coupling components, a thirdsignal representing a ratio between Vyy and Vzz coupling components, afourth signal representing a ratio between Vxx and Vyy couplingcomponents, and a fifth signal representing a ratio between across-coupling sum and a sum of the direct couplings; and determining aformation dip log and a formation anisotropy log based at least in parton said set of signals.
 2. The method of claim 1, further comprisingdetermining a formation resistivity log and a formation strike angle logbased at least in part on said set of signals.
 3. The method of claim 2,further comprising displaying at least one of said logs to a user. 4.The method of claim 1, wherein said determining is performed viainversion by iterative estimation of model parameters.
 5. The method ofclaim 1, wherein said signal measurements represent averagedmeasurements for measurement bins.
 6. The method of claim 5, whereinsaid signal measurements are calibrated, combined, and/or compensated toimprove their reliability.
 7. The method of claim 1, wherein the tool isa logging while drilling tool having one or more tilted antennas.
 8. Themethod of claim 1, wherein the tool is a wireline tool having at leastone triad of orthogonal transmit antennas and at least one triad oforthogonal receive antennas.
 9. The method of claim 1, wherein the firstsignal is representable as S1=Vzz(d1)/Vzz(d2), the second signal isrepresentable as S2=Vxx/Vzz, the third signal is representable asS3=Vyy/Vzz, and the fourth signal is representable as Vxx/Vyy.
 10. Themethod of claim 9, wherein the fifth signal is expressible as$S_{5} = {1 + {\frac{V_{xz} + V_{zx}}{{\frac{1}{2}V_{xx}} + {\frac{1}{2}V_{yy}} + V_{zz}}.}}$11. The method of claim 1, further comprising steering a drill bit basedon the determined formation dip log.
 12. A resistivity logging systemthat comprises: a memory that stores logging software; and at least oneprocessor coupled to the memory to execute the logging software, thesoftware causing the at least one processor to: obtain signalmeasurements collected by an azimuthally sensitive electromagneticlogging tool as a function of position in a borehole, the tool having atleast two spacing distances (d1, d2) between transmit and receiveantennas; derive from the signal measurements: orthogonal directcouplings (Vxx, Vyy, Vzz) and a cross-coupling sum (Vxz+Vzx) or(Vyz+Vzy); generate a set of signals as a function of position in theborehole, the set including: a first signal representing a ratio betweenVzz coupling components at different spacing distances, a second signalrepresenting a ratio between Vxx and Vzz coupling components, a thirdsignal representing a ratio between Vyy and Vzz coupling components, afourth signal representing a ratio between Vxx and Vyy couplingcomponents, and a fifth signal representing a ratio between across-coupling sum and a sum of the direct couplings; and determine aformation dip log and a formation anisotropy log based at least in parton said set of signals.
 13. The system of claim 12, wherein the softwarefurther causes the at least one processor to determine a formationresistivity log and a formation strike angle log based at least in parton said set of signals.
 14. The system of claim 13, wherein the softwarefurther causes the at least one processor to display at least one ofsaid logs to a user.
 15. The system of claim 12, wherein as part of saiddetermining, the software causes the processor to iteratively estimateformation dip and anisotropy until the generated set of signals ismatched by a modeled set of signals.
 16. The system of claim 12, whereinsaid signal measurements represent averaged measurements for measurementbins.
 17. The system of claim 16, wherein said signal measurements arecalibrated, combined, and/or compensated to improve their reliability.18. The system of claim 12, wherein the tool is a logging while drillingtool having one or more tilted antennas.
 19. The system of claim 12,wherein the tool is a wireline tool having at least one triad oforthogonal transmit antennas and at least one triad of orthogonalreceive antennas.
 20. The system of claim 12, wherein the first signalis expressible as S1=Vzz(d1)/Vzz(d2), the second signal is expressibleas S2=Vxx/Vzz, the third signal is expressible as S3=Vyy/Vzz, and thefourth signal is expressible as Vxx/Vyy.
 21. The system of claim 20,wherein the fifth signal is expressible as$S_{5} = {1 + {\frac{V_{xz} + V_{zx}}{{\frac{1}{2}V_{xx}} + {\frac{1}{2}V_{yy}} + V_{zz}}.}}$22. The system of claim 12, wherein a drill bit is steered based on thedetermined formation dip log.